X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?a=blobdiff_plain;ds=sidebyside;f=topics%2F_week5_system_F.mdwn;h=4afb43bae5fd783c4007e9158ab3a15bb44858fd;hb=c09bd7005b179db8ab4c09c4c60be32d1a1c8881;hp=2d9035a7baa521541ecb99268fe1ea539d2ab5d8;hpb=7e490929c94a664464007a6d850e72959d833d60;p=lambda.git diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn index 2d9035a7..4afb43ba 100644 --- a/topics/_week5_system_F.mdwn +++ b/topics/_week5_system_F.mdwn @@ -1,3 +1,5 @@ +[[!toc levels=2]] + # System F and recursive types In the simply-typed lambda calculus, we write types like σ @@ -68,7 +70,7 @@ variables in its body, except that unlike λ, Λ binds type variables instead of expression variables. So in the expression -Λ 'a (λ x:'a . x) +Λ Î± (λ x:α . x) the Λ binds the type variable `'a` that occurs in the λ abstract. Of course, as long as type @@ -77,10 +79,11 @@ tick marks, Grecification, or capitalization), there is no need to distinguish expression abstraction from type abstraction by also changing the shape of the lambda. -This expression is a polymorphic version of the identity function. It -defines one general identity function that can be adapted for use with -expressions of any type. In order to get it ready to apply this -identity function to, say, a variable of type boolean, just do this: +The expression immediately below is a polymorphic version of the +identity function. It defines one general identity function that can +be adapted for use with expressions of any type. In order to get it +ready to apply this identity function to, say, a variable of type +boolean, just do this: (Λ 'a (λ x:'a . x)) [t] @@ -142,13 +145,16 @@ other words, the instantiation turns a Church number into a pair-manipulating function, which is the heart of the strategy for this version of predecessor. -But of course, the type `Pair` (in this simplified example) is defined -in terms of Church numbers. If we tried to replace the type for -Church numbers with a concrete (simple) type, we would have to replace -each `X` with `(N -> N -> N) -> N`. But then we'd have to replace -each `N` with `(X -> X) -> X -> X`. And then replace each `X` -with... ad infinitum. If we had to choose a concrete type built -entirely from explicit base types, we'd be unable to proceed. +Could we try to build a system for doing Church arithmetic in which +the type for numbers always manipulated ordered pairs? The problem is +that the ordered pairs we need here are pairs of numbers. If we tried +to replace the type for Church numbers with a concrete (simple) type, +we would have to replace each `X` with the type for Pairs, `(N -> N -> +N) -> N`. But then we'd have to replace each of these `N`'s with the +type for Church numbers, `(X -> X) -> X -> X`. And then we'd have to +replace each of these `X`'s with... ad infinitum. If we had to choose +a concrete type built entirely from explicit base types, we'd be +unable to proceed. [See Benjamin C. Pierce. 2002. *Types and Programming Languages*, MIT Press, chapter 23.] @@ -194,8 +200,91 @@ be strongly normalizing, from which it follows that System F is not Turing complete. -Types in OCaml --------------- +## Polymorphism in natural language + +Is the simply-typed lambda calclus enough for analyzing natural +language, or do we need polymorphic types? Or something even more expressive? + +The classic case study motivating polymorphism in natural language +comes from coordination. (The locus classicus is Partee and Rooth +1983.) + + Ann left and Bill left. + Ann left and slept. + Ann and Bill left. + Ann read and reviewed the book. + +In English (likewise, many other languages), *and* can coordinate +clauses, verb phrases, determiner phrases, transitive verbs, and many +other phrase types. In a garden-variety simply-typed grammar, each +kind of conjunct has a different semantic type, and so we would need +an independent rule for each one. Yet there is a strong intuition +that the contribution of *and* remains constant across all of these +uses. Can we capture this using polymorphic types? + + Ann, Bill e + left, slept e -> t + read, reviewed e -> e -> t + +With these basic types, we want to say something like this: + + and:t->t->t = lambda l:t . lambda r:t . l r false + and = lambda 'a . lambda 'b . + lambda l:'a->'b . lambda r:'a->'b . + lambda x:'a . and:'b (l x) (r x) + +The idea is that the basic *and* conjoins expressions of type `t`, and +when *and* conjoins functional types, it builds a function that +distributes its argument across the two conjuncts and conjoins the two +results. So `Ann left and slept` will evaluate to `(\x.and(left +x)(slept x)) ann`. Following the terminology of Partee and Rooth, the +strategy of defining the coordination of expressions with complex +types in terms of the coordination of expressions with less complex +types is known as Generalized Coordination. + +But the definitions just given are not well-formed expressions in +System F. There are three problems. The first is that we have two +definitions of the same word. The intention is for one of the +definitions to be operative when the type of its arguments is type +`t`, but we have no way of conditioning evaluation on the *type* of an +argument. The second is that for the polymorphic definition, the term +*and* occurs inside of the definition. System F does not have +recursion. + +The third problem is more subtle. The defintion as given takes two +types as parameters: the type of the first argument expected by each +conjunct, and the type of the result of applying each conjunct to an +argument of that type. We would like to instantiate the recursive use +of *and* in the definition by using the result type. But fully +instantiating the definition as given requires type application to a +pair of types, not to just a single type. We want to somehow +guarantee that 'b will always itself be a complex type. + +So conjunction and disjunction provide a compelling motivation for +polymorphism in natural language, but we don't yet have the ability to +build the polymorphism into a formal system. + +And in fact, discussions of generalized coordination in the +linguistics literature are almost always left as a meta-level +generalizations over a basic simply-typed grammar. For instance, in +Hendriks' 1992:74 dissertation, generalized coordination is +implemented as a method for generating a suitable set of translation +rules, which are in turn expressed in a simply-typed grammar. + +Not incidentally, we're not aware of any programming language that +makes generalized coordination available, despite is naturalness and +ubiquity in natural language. That is, coordination in programming +languages is always at the sentential level. You might be able to +evaluate `(delete file1) and (delete file2)`, but never `delete (file1 +and file2)`. + +We'll return to thinking about generalized coordination as we get +deeper into types. There will be an analysis in term of continuations +that will be particularly satisfying. + + +#Types in OCaml + OCaml has type inference: the system can often infer what the type of an expression must be, based on the type of other known expressions.